Third Week Final Test

The existence of the derivative is sufficient but not a necessity for the proper approximation the function by straight line (e.g. in case of $|x|$ )

If $f'(a)=0$ then there is a neighbourhood of $x=a$ such as $f(x)$ is a constant (non-changing) function in there

The change of argument is in the definition of the derivative is considered by its absolute value, thus it could not be negative

The sum of two non-differentiable at the given point functions could be differentiable

Correct

Correct!

\left(- \log{\left (\sin{\left (x \right )} \right )} \sin{\left (x \right )} + \frac{\cos^{2}{\left (x \right )}}{\sin{\left (x \right )}}\right) \sin^{\cos{\left (x \right )}}{\left (x \right )}

(sin(x))^(cos(x))*((cos(x)^2)/(sin(x))-(sin(x))*ln(sin(x)))

Correct

e

E

Correct

123.52

123.52

Correct

In the point of maximum one can always find left neighbourhood with only positive derivative and right neighbourhood with only negative derivative

The function could remain convex before and after the point where $f''(x)=0$

Correct

Correct!

The existence of the derivative is a necessity for the extremum

If the function changes it monotonicity in the given point, then it is an extremum

Correct

Correct!

2;CONVEX

Correct

1;MAXIMUM

Correct